Roots and critical points of polynomials over Cayley–Dickson algebras

Abstract

Over a composition algebraWe examine whether this is true for general Cayley-Dickson algebras. The conclusion is that it is when f (x) is linear or monic quadratic, but it is false in general. Similar questions about the connections between f and its companion C f (x) = f (x) • f (x) are studied. Finally, we compute the left eigenvalues of 2 × 2 octonion matrices.

Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras

“…Recall that when dim A ≤ 8, every root of f (x) is a root of C f (x), and every root of C f (x) has an element quadratically equivalent to it. In [5] it was pointed out that f (x) = αx ∈ S[x] has β as a root even though β is not a root of C f (x) = 2x 2 (where α and β are as in Example 2.3). The second part of the property stated in the first line of this paragraph has not been dealt with though, which brings up the following question.…”

“…In [11] it was shown that this does not extend to f (x) ∈ H[x] (here the derivative f ′ (x) is defined formally). In [5,Theorem 4.2] it was proven that the spherical critical points of f (x) (again, f ′ (x) is defined formally) are contained in the convex hull of the roots of C f (x) for any f (x) ∈ A[x] where A is any real anisotropic Cayley-Dickson algebra. Question 2.13.…”

“…In this article we address several questions that came up in the writing of the articles [5,7]. In Section 2 we discuss the decomposition of the form f (x) = g(x) • (x − λ) for roots λ of polynomials with coefficients in a Cayley-Dickson algebra.…”

Alternating Roots of Polynomials over Cayley-Dickson Algebras

“…A detailed description of how to find the roots of a standard polynomial over H appeared in [4], and a generalization for arbitrary quaternion division algebras appeared in [3]. In [1], a description of the roots of a standard octonion polynomial was provided, and spherical roots of polynomials over arbitrary Cayley-Dickson algebras were studied in [2].…”